With completely analogous arguments as in the proof of
Theorem REF  we find
with Appendix  that any
\(g \in \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  for which \(g_1\)  is invertible
has a unique lift \(\pi _a(g) \in \mathop {\rm U{}}\nolimits (\Lambda (\mathcal {H}))\)  with
\( \pi _a(g) \Omega = c(g) e^{-\widehat{T}(g)} \quad \mbox{ for } \quad T(g) := g_2 g_1^{-1} \in \mathop {{\rm Aherm}}\nolimits (\mathcal {H})_a,c(g) = \Vert e^{-\widehat{T}(g)}\Vert ^{-1} \) 
(cf. Lemma REF ).
This implies that \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  is a Lie group,
its representation \(\pi _a\)  on \(\Lambda (\mathcal {H})\)  is smooth, and
that \(\Omega \)  is a smooth vector.
Since \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  acts smoothly on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) ,
the space \(\Lambda (\mathcal {H})^\infty \)  of smooth vectors is invariant under
\(\pi _0(\mathop {{\rm CAR}}\nolimits (\mathcal {H}))\) , so that the irreducibility of the
representation of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  on \(\Lambda (\mathcal {H})\) 
implies the smoothness of \(\pi _a\) .
For the Lie algebra cocycle defining \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) ,
we find as in the proof of Theorem REF 
with Lemma REF (iii)
\(\eta (x,y)= 2 \mathop {{\rm Im}}\nolimits \langle \widehat{x}_2, \widehat{y}_2 \rangle = -\mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits (x_2y_2) = -\frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x_2,y_2]).\) 

\(\Box \)
Remark 10.3   As in Subsection REF ,
we obtain a Banach–Lie group
\(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})&:= \Big \lbrace  g = g_1 + g_2 \in \mathop {\rm O{}}\nolimits (\mathcal {H})\colon \Vert g_2 \Vert _2 < \infty , \Vert \mathbf {1} - g_1\Vert _1 < \infty \rbrace \) 

with
\( \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}) \cong (\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/N,\quad N \cong \mathop {\rm U{}}\nolimits (\mathcal {H}) \cap \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H}) = \mathop {\rm U{}}\nolimits _1(\mathcal {H}) \) 
(cf. [1]}). From [1]}
we recall that \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  has two connected components
and that its fundamental group is trivial.
On the other hand
\( \pi _0(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \cong {\mathbb {Z}}/2{\mathbb {Z}}\cong \pi _1(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \) 
([3]}, [1]}),
so that there exists a simply connected 2-fold covering group
\(q \colon  {\rm Spin}_{1,2}(\mathcal {H})\rightarrow \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})_0\) 
(cf. [5]} for a discussion of the smaller group
\({\rm Spin}_1(\mathcal {H})\)  covering \(\mathop {{\rm SO}}\nolimits _1(\mathcal {H}) = \mathop {\rm O{}}\nolimits _1(\mathcal {H})_0\) ).
As the
inclusion \(\mathop {\rm U{}}\nolimits _{1}(\mathcal {H}) \rightarrow \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})\)  induces a surjective homomorphism
\( \pi _1(\mathop {\rm U{}}\nolimits _1(\mathcal {H})) \cong {\mathbb {Z}}\rightarrow \pi _1(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \cong {\mathbb {Z}}/2, \) 
\(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) := q^{-1}(\mathop {\rm U{}}\nolimits _1(\mathcal {H}))\)  is the unique
2-fold connected covering of
\(\mathop {\rm U{}}\nolimits _1(\mathcal {H})\)  from Subsection REF .
We further have an embedding
\(\sigma \colon  {\mathfrak {o}}_{1,2}(\mathcal {H}) \rightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H}), \quad \sigma (x) := \Big (-\frac{1}{2i} \mathop {{\rm tr}}\nolimits (x_1), x\Big )\) 
of Banach–Lie algebras.
On the subgroup \(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \subseteq {\rm Spin}_{1,2}(\mathcal {H})\) ,
\(\sigma \)  integrates to a group homomorphism
\(\sigma _G(g) := (g, \sqrt{\det }(g)^{-1}\pi _s(g))\) 
(cf. Subsection REF )
and \(\sigma \)  integrates to a morphism of Banach–Lie groups
\(\sigma _G \colon  {\rm Spin}_{1,2}(\mathcal {H}) \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H}).\) 
Combining this map with the canonical inclusion
\(\mathop {\rm U{}}\nolimits (\mathcal {H}) \hookrightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) , the equivariance of
\(\sigma _G\)  under conjugation with unitary operators implies the
existence of a homomorphism
\( \mu \colon  {\rm Spin}_{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H}),\quad (g,u) \mapsto \sigma _G(g)u.\) 

The following proposition is proved as Proposition REF ,
using the representation \(\pi _a\)  instead of \(\pi _s\) .
Proposition 10.4  The homomorphism \(\mu \)  factors through an
isomorphism
\( \overline{\mu }\colon  ({\rm Spin}_{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/\ker \mu \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\) 
of connected Banach–Lie groups with
\(\ker \mu \cong \mathop {{\rm SU}}\nolimits _1(\mathcal {H}).\)

Remark 10.5  
The embedding \({\mathfrak {o}}_{1,2}(\mathcal {H}) \hookrightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) 
restricts in particular to an embedding \({\mathfrak {o}}_1(\mathcal {H}) \hookrightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) . One can show that the operators
\({\tt d}\pi _a(x)\)  are bounded for \(x \in {\mathfrak {o}}_1(\mathcal {H})\) 
(cf. [6]}, [7]}), so that we obtain
a morphism of Banach–Lie algebras
\( {\mathfrak {o}}_1(\mathcal {H}) \rightarrow {\mathfrak {u}}(\Lambda (\mathcal {H})), \quad x \mapsto {\tt d}\pi _a(x) - \frac{1}{2} \mathop {{\rm tr}}\nolimits (x_1)\mathbf {1}. \) 

Definition 10.6  To determine the open cone \(W_{\pi _a}\) 
for the spin representation, we have to take a closer
look at natural cones in \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) .
Let \({\mathfrak {u}}_\infty (\mathcal {H})\)  denote the ideal of compact skew-hermitian
operators in \({\mathfrak {u}}(\mathcal {H})\)  and recall the Calkin algebra
\(\mathop {{\rm Cal}}\nolimits (\mathcal {H}) := B(\mathcal {H})/B_\infty (\mathcal {H})\) , where
\(B_\infty (\mathcal {H})\)  denotes the ideal of compact operators
in the \(C^*\) -algebra \(B(\mathcal {H})\) . Then the surjection
\(B(\mathcal {H}) \rightarrow \mathop {{\rm Cal}}\nolimits (B(\mathcal {H}))\)  induces an isomorphism
\( {\mathfrak {u}}(\mathcal {H})/{\mathfrak {u}}_\infty (\mathcal {H}) \cong {\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H})). \) 
Next we note that
\(\lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  x^* - x \in {\mathfrak {u}}_\infty (\mathcal {H})\rbrace \) 
is a closed ideal of \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) 
which defines a quotient morphism
\(q \colon  {\mathfrak {o}}_{\rm res}(\mathcal {H}) \rightarrow {\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H})).\) 
We thus obtain an open invariant cone in \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) 
by
\( W_{{\mathfrak {o}}_{\rm res}(\mathcal {H})} := q^{-1}(C_{{\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H}))})= \lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  -i q(x) <\!\!< 0 \rbrace  \) 
(cf. Example REF ). We write
\(W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})}\)  for its inverse image in the
central extension \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) .

Theorem 10.7  
The spin representation \((\pi _a, \Lambda (\mathcal {H}))\)  of
the connected Lie group \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  is semibounded
with \(W_{\pi _a} = W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})}\) .

